Chronic Bronchitis: Given the patient's age and chronic cough, chronic bronchitis is a likely cause of patches in the right lower lobe of the lung. Chronic bronchitis is a type of chronic obstructive pulmonary disease (COPD) that can cause inflammation and scarring in the lungs, leading to patches on imaging studies.

Pneumonia: A common cause of lung patches, pneumonia can be caused by bacteria, viruses, or fungi and can present with cough, fever, and difficulty breathing.
Lung Cancer: As the patient is 65 years old, lung cancer is a possible cause of lung patches, especially if the patient has a history of smoking.
Tuberculosis (TB): Although less common in developed countries, TB can cause lung patches and should be considered, especially if the patient has a history of exposure or travel to endemic areas.
Pulmonary Fibrosis: A condition in which lung tissue becomes scarred, pulmonary fibrosis can cause lung patches and can be idiopathic or secondary to other conditions such as rheumatoid arthritis or scleroderma.

Pulmonary Embolism (PE): Although less likely to cause patches, PE can present with cough and should be considered, especially if the patient has risk factors such as immobility, cancer, or recent surgery.
Empyema: A collection of pus in the pleural space, empyema can cause lung patches and should be considered if the patient has a history of pneumonia or recent chest surgery.
Lung Abscess: A cavity filled with pus in the lung, lung abscess can cause lung patches and should be considered if the patient has a history of aspiration or immunocompromised state.

Sarcoidosis: A condition in which granulomas form in the lungs, sarcoidosis can cause lung patches and should be considered if the patient has a history of skin or eye symptoms.
Lymphangitic Carcinomatosis: A rare condition in which cancer spreads to the lymphatic vessels in the lungs, lymphangitic carcinomatosis can cause lung patches and should be considered if the patient has a history of cancer.
Eosinophilic Pneumonia: A rare condition in which eosinophils accumulate in the lungs, eosinophilic pneumonia can cause lung patches and should be considered if the patient has a history of asthma or parasitic infection.

The management strategy will depend on the underlying diagnosis. However, initial steps may include:

Chest CT scan: To further evaluate the lung patches and guide further management.
Pulmonary function tests (PFTs): To assess lung function and guide further management.
Bronchoscopy: To obtain samples for culture and cytology.
Antibiotics: If pneumonia or other infectious causes are suspected.
Oxygen therapy: If the patient is hypoxemic.
Referral to a pulmonologist: For further evaluation and management.